This will take the form

y=c (cos cos +sin 0 sin ) =c cos (0—4),

These two last equations give, by squaring and adding, c2=a2+b2+2 a b cos o,

Hence and can always be determined in terms of the given quantities a, b, and o, and y can thus be reduced to the form

where c and are independent both of x and t. This last equation evidently denotes a simple harmonic undulation of wave length a, velocity v and amplitude c.

56. On the supposition of a slight difference in the wave-lengths of the two components, the circumstances are approximately represented by making vary slowly with t. Then the greatest value of c2 will be obtained by putting =0, and will be a2+b2 + 2 a b or (a + b)2; and the least will be obtained by putting =7, and will be (a-6)2. The mean value of c2 will (as explained in § 35) be a2+b2.

57. The case discussed in §§ 54-56 is that which occurs when two simple tones of nearly the same pitch are emitted from two neighbouring sources of sound. The waves from the two sources are propagated through the surrounding air with the same velocity, and those which belong to the sound of higher pitch are slightly shorter than the others.

58. Next, let the directions of propagation be opposite for the two component undulations, their wavelengths being exactly equal, and also their amplitudes.

First, suppose the vibrations to be transverse, Then the successive actions will be understood by inspection of the annexed figures 21, 22, 23, 24. The heavy curve in each figure represents a definite portion A B C D A' of the vibrating string, and the two lighter curves above it represent the two component undulations travelling in opposite directions, as suggested by the arrows. The interval of time from each figure to the next is a quarter period.

In Fig. 21 there is a coincidence of crests at в and of troughs at D.

In Fig. 22 a crest coincides with a trough at A, C, and A'.

In Fig. 23 there is coincidence of troughs at в and of crests at D.

In Fig. 24 there is a coincidence of crest with trough at A, C, and A'. After another quarter period the state of things in Fig. 21 will recur.

E

In each quarter period one component has travelled a quarter wave-length to the left, and the other the same distance to the right, so that their displacement relative to each other is half a wave-length. Comparing together the four positions of the string

here represented, it is easy to understand that the points A, C, and A' remain permanently at rest, and the points B and D midway between them undergo the largest displacements. On the other hand, if we attend to the direction of a tangent at a given point of the string, we see that this direction changes most at A, C, and A', and

The points of perma

does not change at all at B and D. nent rest, A, C, and a' are called nodes, and the points of maximum displacement, B and D, antinodes.

59. Now, passing to the case of longitudinal vibrations, consider the state of things at the moment when the maximum compressions in the two sets of waves are in coincidence, and, therefore, the maximum extensions also in coincidence. At this moment the effect of one set of waves, as regards compression or extension, is at every point the same as the effect of the other set, both in kind and in degree, and the resultant effect will be double of either.

But as regards velocity the effects destroy one another; for at the places of compression the velocity due to either set is in the direction of propagation of that set, and these two directions of propagation are opposite; while at the places of extension the velocities are opposite to the directions of propagation, and, therefore, are here also opposite to each other. Hence, there is instantaneous rest all along the waves.

Next, consider the state of things at the moment when the maximum compressions of one set coincide with the maximum extensions of the other. We shall then have mutual destruction of effect as regards compression and extension, but the velocities due to the two sets will be identical, and therefore the resultant velocity will be double of either.

The time intervening between the two moments herc

considered is only a quarter of a period; for since the two sets of waves are travelling opposite ways, their relative velocity is double the absolute velocity of either; and the interval of time in question is the time of travelling over half a wave-length with this double velocity, or is the time in which each set of waves advances a quarter of a wave-length.

60. After another quarter period the waves will have advanced another quarter of a wave-length, and we shall again have double compressions and extensions, with instantaneous rest everywhere; but as the waves have advanced half a wave-length since the moment first considered, the places of compression and extension are interchanged. The points which were then the places of maximum compression are now the places of maximum extension, and vice versa. Hence there are certain points, situated at regular intervals of half a wave-length from each other, which are alternately the points of maximum compression and of maximum extension. At these points the compression or extension is constantly double of that due to either set of waves separately, and the velocity is constantly zero-in other words, these are places of permanent rest. They are called nodes.

61. When the maximum compressions of one set are in coincidence with the maximum extensions of the other, the points at which these coincidences occur are midway between the nodes. At these points (which are called antinodes) the velocities at the moment now